Question

which graph represents the equation y = -3x ?

Explanation:

Step1: Analyze the equation form

The equation \( y = -3x \) is in slope - intercept form \( y=mx + b \), where \( m=-3 \) (slope) and \( b = 0 \) (y - intercept). So the line should pass through the origin \((0,0)\) and have a negative slope (since \( m=-3<0 \)), meaning it goes down from left to right.

Step2: Analyze each graph

• First graph: The line has a positive slope (goes up from left to right), so it can't be \( y=-3x \).
• Second graph: The line passes through the origin? Wait, no, let's check the slope. Wait, actually, let's check the key points. For \( y = - 3x \), when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y=-3 \); when \( x=- 1 \), \( y = 3 \). The second graph: Let's see the direction. It has a negative slope (goes down from left to right) and passes through points that would fit \( y=-3x \)? Wait, no, wait the fourth graph? Wait, no, let's re - check. Wait the equation is \( y=-3x \), slope is - 3, which is a steep negative slope. Let's check the graphs again. The second graph: Let's assume the grids are 1 unit per square. For \( y=-3x \), when \( x = 1 \), \( y=-3 \); when \( x=-1 \), \( y = 3 \). The second graph: if we take a point, say when \( x=-2 \), \( y = 6 \)? No, maybe I misread. Wait the fourth graph? Wait no, the second graph: let's see the slope. The slope \( m=\frac{\Delta y}{\Delta x}\). If a line goes from, say, \( (x_1,y_1)\) to \( (x_2,y_2) \), \( m=\frac{y_2 - y_1}{x_2 - x_1}\). For \( y=-3x \), \( m=-3 \). Let's check the second graph: suppose two points on it, if it goes from \( (x=-2,y = 6) \) to \( (x=-1,y = 3) \), then \( m=\frac{3 - 6}{-1-(-2)}=\frac{-3}{1}=-3 \), which matches. Wait, but also, the line should pass through the origin? Wait \( y=-3x \) when \( x = 0 \), \( y = 0 \), so it passes through the origin. Let's check the graphs again. The fourth graph: does it pass through the origin? The second graph: let's see, if the line in the second graph passes through \( (x=-1,y = 3) \) and \( (x = 0,y = 0) \) and \( (x = 1,y=-3) \), then it has slope - 3 and passes through the origin. Wait, maybe I made a mistake earlier. Wait the first graph: positive slope, third graph: positive slope, fourth graph: let's check slope. If the fourth graph has a line that goes from, say, \( (x = 0,y = 0) \) to \( (x = 1,y=-3) \)? Wait no, the fourth graph's line: let's see the direction. Wait the second graph: the line is going down from left to right (negative slope) and passes through the origin? Wait, maybe the second graph is the one. Wait, no, let's re - express. The equation \( y=-3x \) is a linear function with slope - 3 and y - intercept 0. So the graph should be a straight line through the origin with a steep negative slope (going down 3 units for every 1 unit to the right). Among the four graphs, the second graph (the one with the line going down from left to right, passing through points that satisfy \( y=-3x \)) is the correct one. Wait, maybe I messed up the numbering. Wait the original graphs: first (leftmost) has positive slope, second: negative slope, third: positive slope, fourth: negative slope but maybe different slope. Wait the slope of \( y=-3x \) is - 3, which is a steep slope. So the second graph: let's check the slope. If we take two points on the second graph, say \( (x=-1,y = 3) \) and \( (x = 0,y = 0) \), the slope is \( \frac{0 - 3}{0-(-1)}=-3 \), which matches. So the second graph (the one with the line going down from left to right, passing through the origin - like points) is the correct graph.

Answer:

The second graph (the one with the line that has a negative slope and passes through the origin - related points)